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## G = (C2×C10)⋊D8order 320 = 26·5

### 3rd semidirect product of C2×C10 and D8 acting via D8/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — (C2×C10)⋊D8
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C20⋊7D4 — (C2×C10)⋊D8
 Lower central C5 — C10 — C2×C20 — (C2×C10)⋊D8
 Upper central C1 — C22 — C22×C4 — C4⋊D4

Generators and relations for (C2×C10)⋊D8
G = < a,b,c,d | a2=b10=c8=d2=1, ab=ba, ac=ca, dad=ab5, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 550 in 134 conjugacy classes, 45 normal (39 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×3], C22, C22 [×2], C22 [×8], C5, C8 [×3], C2×C4 [×2], C2×C4 [×4], D4 [×8], C23, C23 [×2], D5, C10 [×3], C10 [×3], C22⋊C4 [×2], C4⋊C4, C4⋊C4, C2×C8 [×4], D8 [×2], C22×C4, C2×D4, C2×D4 [×3], Dic5, C20 [×2], C20 [×2], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×5], D4⋊C4 [×2], C2.D8, C4⋊D4, C4⋊D4, C22×C8, C2×D8, C52C8 [×2], C52C8, D20 [×2], C2×Dic5, C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×3], C5×D4 [×4], C22×D5, C22×C10, C22×C10, C87D4, C2×C52C8 [×2], C2×C52C8 [×2], C4⋊Dic5, D10⋊C4, D4⋊D5 [×2], C5×C22⋊C4, C5×C4⋊C4, C2×D20, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, C10.D8, D206C4, D4⋊Dic5, C22×C52C8, C207D4, C2×D4⋊D5, C5×C4⋊D4, (C2×C10)⋊D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, D8 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×D8, C4○D8, C5⋊D4 [×2], C22×D5, C87D4, D4⋊D5 [×2], D4×D5, D42D5, C2×C5⋊D4, C2×D4⋊D5, Dic5⋊D4, D4.8D10, (C2×C10)⋊D8

Smallest permutation representation of (C2×C10)⋊D8
On 160 points
Generators in S160
(1 98)(2 99)(3 100)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 97)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 81)(18 82)(19 83)(20 84)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110)(111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160)
(1 58 38 90 30 70 50 78)(2 57 39 89 21 69 41 77)(3 56 40 88 22 68 42 76)(4 55 31 87 23 67 43 75)(5 54 32 86 24 66 44 74)(6 53 33 85 25 65 45 73)(7 52 34 84 26 64 46 72)(8 51 35 83 27 63 47 71)(9 60 36 82 28 62 48 80)(10 59 37 81 29 61 49 79)(11 105 145 125 153 93 133 113)(12 104 146 124 154 92 134 112)(13 103 147 123 155 91 135 111)(14 102 148 122 156 100 136 120)(15 101 149 121 157 99 137 119)(16 110 150 130 158 98 138 118)(17 109 141 129 159 97 139 117)(18 108 142 128 160 96 140 116)(19 107 143 127 151 95 131 115)(20 106 144 126 152 94 132 114)
(2 10)(3 9)(4 8)(5 7)(11 150)(12 149)(13 148)(14 147)(15 146)(16 145)(17 144)(18 143)(19 142)(20 141)(21 29)(22 28)(23 27)(24 26)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 50)(39 49)(40 48)(51 75)(52 74)(53 73)(54 72)(55 71)(56 80)(57 79)(58 78)(59 77)(60 76)(61 89)(62 88)(63 87)(64 86)(65 85)(66 84)(67 83)(68 82)(69 81)(70 90)(91 100)(92 99)(93 98)(94 97)(95 96)(101 104)(102 103)(105 110)(106 109)(107 108)(111 122)(112 121)(113 130)(114 129)(115 128)(116 127)(117 126)(118 125)(119 124)(120 123)(131 160)(132 159)(133 158)(134 157)(135 156)(136 155)(137 154)(138 153)(139 152)(140 151)

G:=sub<Sym(160)| (1,98)(2,99)(3,100)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,81)(18,82)(19,83)(20,84)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,58,38,90,30,70,50,78)(2,57,39,89,21,69,41,77)(3,56,40,88,22,68,42,76)(4,55,31,87,23,67,43,75)(5,54,32,86,24,66,44,74)(6,53,33,85,25,65,45,73)(7,52,34,84,26,64,46,72)(8,51,35,83,27,63,47,71)(9,60,36,82,28,62,48,80)(10,59,37,81,29,61,49,79)(11,105,145,125,153,93,133,113)(12,104,146,124,154,92,134,112)(13,103,147,123,155,91,135,111)(14,102,148,122,156,100,136,120)(15,101,149,121,157,99,137,119)(16,110,150,130,158,98,138,118)(17,109,141,129,159,97,139,117)(18,108,142,128,160,96,140,116)(19,107,143,127,151,95,131,115)(20,106,144,126,152,94,132,114), (2,10)(3,9)(4,8)(5,7)(11,150)(12,149)(13,148)(14,147)(15,146)(16,145)(17,144)(18,143)(19,142)(20,141)(21,29)(22,28)(23,27)(24,26)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,50)(39,49)(40,48)(51,75)(52,74)(53,73)(54,72)(55,71)(56,80)(57,79)(58,78)(59,77)(60,76)(61,89)(62,88)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,90)(91,100)(92,99)(93,98)(94,97)(95,96)(101,104)(102,103)(105,110)(106,109)(107,108)(111,122)(112,121)(113,130)(114,129)(115,128)(116,127)(117,126)(118,125)(119,124)(120,123)(131,160)(132,159)(133,158)(134,157)(135,156)(136,155)(137,154)(138,153)(139,152)(140,151)>;

G:=Group( (1,98)(2,99)(3,100)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,81)(18,82)(19,83)(20,84)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,58,38,90,30,70,50,78)(2,57,39,89,21,69,41,77)(3,56,40,88,22,68,42,76)(4,55,31,87,23,67,43,75)(5,54,32,86,24,66,44,74)(6,53,33,85,25,65,45,73)(7,52,34,84,26,64,46,72)(8,51,35,83,27,63,47,71)(9,60,36,82,28,62,48,80)(10,59,37,81,29,61,49,79)(11,105,145,125,153,93,133,113)(12,104,146,124,154,92,134,112)(13,103,147,123,155,91,135,111)(14,102,148,122,156,100,136,120)(15,101,149,121,157,99,137,119)(16,110,150,130,158,98,138,118)(17,109,141,129,159,97,139,117)(18,108,142,128,160,96,140,116)(19,107,143,127,151,95,131,115)(20,106,144,126,152,94,132,114), (2,10)(3,9)(4,8)(5,7)(11,150)(12,149)(13,148)(14,147)(15,146)(16,145)(17,144)(18,143)(19,142)(20,141)(21,29)(22,28)(23,27)(24,26)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,50)(39,49)(40,48)(51,75)(52,74)(53,73)(54,72)(55,71)(56,80)(57,79)(58,78)(59,77)(60,76)(61,89)(62,88)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,90)(91,100)(92,99)(93,98)(94,97)(95,96)(101,104)(102,103)(105,110)(106,109)(107,108)(111,122)(112,121)(113,130)(114,129)(115,128)(116,127)(117,126)(118,125)(119,124)(120,123)(131,160)(132,159)(133,158)(134,157)(135,156)(136,155)(137,154)(138,153)(139,152)(140,151) );

G=PermutationGroup([(1,98),(2,99),(3,100),(4,91),(5,92),(6,93),(7,94),(8,95),(9,96),(10,97),(11,85),(12,86),(13,87),(14,88),(15,89),(16,90),(17,81),(18,82),(19,83),(20,84),(21,101),(22,102),(23,103),(24,104),(25,105),(26,106),(27,107),(28,108),(29,109),(30,110),(31,111),(32,112),(33,113),(34,114),(35,115),(36,116),(37,117),(38,118),(39,119),(40,120),(41,121),(42,122),(43,123),(44,124),(45,125),(46,126),(47,127),(48,128),(49,129),(50,130),(51,131),(52,132),(53,133),(54,134),(55,135),(56,136),(57,137),(58,138),(59,139),(60,140),(61,141),(62,142),(63,143),(64,144),(65,145),(66,146),(67,147),(68,148),(69,149),(70,150),(71,151),(72,152),(73,153),(74,154),(75,155),(76,156),(77,157),(78,158),(79,159),(80,160)], [(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110),(111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160)], [(1,58,38,90,30,70,50,78),(2,57,39,89,21,69,41,77),(3,56,40,88,22,68,42,76),(4,55,31,87,23,67,43,75),(5,54,32,86,24,66,44,74),(6,53,33,85,25,65,45,73),(7,52,34,84,26,64,46,72),(8,51,35,83,27,63,47,71),(9,60,36,82,28,62,48,80),(10,59,37,81,29,61,49,79),(11,105,145,125,153,93,133,113),(12,104,146,124,154,92,134,112),(13,103,147,123,155,91,135,111),(14,102,148,122,156,100,136,120),(15,101,149,121,157,99,137,119),(16,110,150,130,158,98,138,118),(17,109,141,129,159,97,139,117),(18,108,142,128,160,96,140,116),(19,107,143,127,151,95,131,115),(20,106,144,126,152,94,132,114)], [(2,10),(3,9),(4,8),(5,7),(11,150),(12,149),(13,148),(14,147),(15,146),(16,145),(17,144),(18,143),(19,142),(20,141),(21,29),(22,28),(23,27),(24,26),(31,47),(32,46),(33,45),(34,44),(35,43),(36,42),(37,41),(38,50),(39,49),(40,48),(51,75),(52,74),(53,73),(54,72),(55,71),(56,80),(57,79),(58,78),(59,77),(60,76),(61,89),(62,88),(63,87),(64,86),(65,85),(66,84),(67,83),(68,82),(69,81),(70,90),(91,100),(92,99),(93,98),(94,97),(95,96),(101,104),(102,103),(105,110),(106,109),(107,108),(111,122),(112,121),(113,130),(114,129),(115,128),(116,127),(117,126),(118,125),(119,124),(120,123),(131,160),(132,159),(133,158),(134,157),(135,156),(136,155),(137,154),(138,153),(139,152),(140,151)])

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 5A 5B 8A ··· 8H 10A ··· 10F 10G 10H 10I 10J 10K 10L 10M 10N 20A ··· 20H 20I 20J 20K 20L order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 5 5 8 ··· 8 10 ··· 10 10 10 10 10 10 10 10 10 20 ··· 20 20 20 20 20 size 1 1 1 1 2 2 8 40 2 2 2 2 8 40 2 2 10 ··· 10 2 ··· 2 4 4 4 4 8 8 8 8 4 ··· 4 8 8 8 8

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 C4○D4 D8 D10 D10 D10 C4○D8 C5⋊D4 C5⋊D4 D4×D5 D4⋊2D5 D4⋊D5 D4.8D10 kernel (C2×C10)⋊D8 C10.D8 D20⋊6C4 D4⋊Dic5 C22×C5⋊2C8 C20⋊7D4 C2×D4⋊D5 C5×C4⋊D4 C5⋊2C8 C2×C20 C22×C10 C4⋊D4 C20 C2×C10 C4⋊C4 C22×C4 C2×D4 C10 C2×C4 C23 C4 C4 C22 C2 # reps 1 1 1 1 1 1 1 1 2 1 1 2 2 4 2 2 2 4 4 4 2 2 4 4

Matrix representation of (C2×C10)⋊D8 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 9 0 0 0 0 32 0
,
 34 35 0 0 0 0 7 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 34 35 0 0 0 0 8 7 0 0 0 0 0 0 0 7 0 0 0 0 35 24 0 0 0 0 0 0 29 12 0 0 0 0 29 29
,
 34 35 0 0 0 0 8 7 0 0 0 0 0 0 1 0 0 0 0 0 21 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,9,0],[34,7,0,0,0,0,35,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[34,8,0,0,0,0,35,7,0,0,0,0,0,0,0,35,0,0,0,0,7,24,0,0,0,0,0,0,29,29,0,0,0,0,12,29],[34,8,0,0,0,0,35,7,0,0,0,0,0,0,1,21,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;

(C2×C10)⋊D8 in GAP, Magma, Sage, TeX

(C_2\times C_{10})\rtimes D_8
% in TeX

G:=Group("(C2xC10):D8");
// GroupNames label

G:=SmallGroup(320,665);
// by ID

G=gap.SmallGroup(320,665);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,219,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=c^8=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*b^5,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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